Except for the January 25 seminar, Wednesday seminars are 3:45 - 4:35 and Friday seminars are 2:30

Title: Ladder systems and a topological application

Abstract: Ladder systems are defined on certain well-ordered sets, including the set \omega_1, which can be abstractly characterized as the unique (up to a unique order-isomorphism) uncountable well-ordered set such that the set of predecessors of any element is countable. [Compare the characterization of the well-ordered set of natural numbers as an infinite well-ordered set such that every element has only finitely many predecessors.]

A ladder at an element x of a well-ordered set is a strictly ascending sequence whose supremum is x. We use ladder systems on the set \omega_1 to construct some topological spaces and to give a topological application of the Pressing-Down Lemma (also known as Fodor's lemma).

Speaker: Peter Nyikos

Title: Three related topological games

Abstract: A topological game is a game played on a topological space with infinitely many moves made by two players who have opposing objectives. This seminar continues the exposition of a trio of closely related topological games. These were introduced last semester but it is not necessary to have been at any of last semester's seminars to understand this lecture. Two of the games seem different, but the same topological spaces yield the same winning strategies for the corresponding players. The third looks like a minor variation on the second, but it is not known whether the same topological spaces give winning strategies on the second and third games.

Speaker: Peter Nyikos

Title: Ladder systems, coherent families of functions, and numerous equivalent axioms

Abstract: Ladder systems are defined on certain well-ordered sets, including the set \omega_1, which can be abstractly characterized as the unique (up to a unique order-isomorphism) uncountable well-ordered set such that the set of predecessors of any element is countable. [Compare the characterization of the well-ordered set of natural numbers as an infinite well-ordered set such that every element has only finitely many predecessors.]

A ladder at an element x of a well-ordered set is a strictly ascending sequence whose supremum is x. We explore connections between ladder systems and families of functions from initial segments of \omega_1 to the natural numbers. An especially interesting concept is that of coherent families. These have the feature that each function in the familiy differs from earlier ones in only finitely many places.

There will be a handout covering the definitions and the proof of a theorem linking these concepts. No previous exposure to any of the concepts, including \omega_1, is assumed.

Speaker: Peter Nyikos

Title: Three related topological games, continued

Abstract: A topological game is a game played on a topological space with infinitely many moves made by two players who have opposing objectives. This seminar continues the exposition of a trio of closely related topological games. A handout will be given that explains what was covered in earlier seminars and is used in this one.

Speaker: Peter Nyikos

Title: Coherent families of functioins and Aronszajn trees

Abstract: Coherent families of functions have the feature that each function in the familiy differs from earlier ones in only finitely many places. A remarkable application of the axiom of choice is that there is a coherent family of one-to-one functions to the positive integers in which the domains form a well-ordered, uncountable sequence, each domain adding infinitely many elements to each of the earlier domains. The two concepts, one-to-one on the one hand, and an uncountable chain of domains, work against each other, but despite this "instability," the sequence can be uncountable.

Coherent families give a reasonably simple construction of an Aronszajn tree, which is quite difficult to construct otherwise.

Concepts such as "Aronszajn tree" will be explained both in seminar and a handout, which will also cover any necessary information from earlier seminars. No previous exposure to any of the concepts in this lecture is assumed.